Shape optimization problems for functionals with a boundary integral

TL;DR

This paper investigates shape optimization problems involving boundary integrals, establishing the existence of optimal domains under broad conditions and analyzing cases with open sets and finite perimeters, especially for elliptic equations with Robin boundary conditions.

Contribution

It provides new existence results for shape optimization problems with boundary terms, including cases with elliptic PDEs and Robin boundary conditions.

Findings

Existence of optimal domains under general assumptions

Optimal domains can be open sets with finite perimeter

Applicable to elliptic equations with Robin boundary conditions

Abstract

We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin condition on the free boundary. We show the existence of an optimal domain under rather general assumptions and we study the cases when the optimal domains are open sets and have a finite perimeter.